Gemara

46b - Rav Yehudah brei d'Rav Shmuel: Rav says that the eighth/seventh day (i.e. in Chutz la'Aretz we are in Safek if it is the seventh or eighth day) is like the seventh day regarding Sukah, but it is like the eighth day regarding Berachah (this will be explained). R. Yochanan says that it is like the eighth day in both respects.

Version #1: All agree that we sit in the Sukah on this day. They argue about whether or not we bless (Leshev ba'Sukah). Rav says that it is like the seventh day (we bless), and R. Yochanan says that it is like the eighth (we do not bless. They agree that it is like the eighth day regarding Berachah, i.e. in Shemoneh Esre and Birkas ha'Mazon we call it Shemini Atzeres.)

Rav Yosef: Presumably, the Halachah follows R. Yochanan, for Rav Huna bar Bizna and all the Gedolei ha'Dor sat in the Sukah on the eighth/seventh day but did not bless.

Question: Perhaps they hold like the opinion that one blesses only on the first day!

Answer: We have a tradition that they came from the pasture (they had not sat in a Sukah the entire Chag).

Version #2: All agree that we do not bless. They argue about whether or not we sit in the Sukah. Rav says that we sit in it, and R. Yochanan disagrees.

Rav Yosef: Presumably, the Halachah follows R. Yochanan, for Rav Yehudah brei d'Rav Shmuel taught the opinions, and he sat outside on the eighth/seventh day. (end of Version #2)

The Halachah is that we sit in the Sukah, but do not bless.

Rishonim

Rif and Rosh (4:5): The Halachah is that we sit in the Sukah, but do not bless. Since it is Shemini Atzeres, it is a contradiction to bless (this would show that it is the seventh day). We are stringent to sit in the Sukah and conduct like Yom Tov.

Ran (DH v'Hilchesa): Even though nowadays the calendar is fixed (and we have no Safek about the day), we must guard the custom of our ancestors. The Rif holds that we do not bless lest people disgrace Yom Tov. We count the Omer on the second night of Pesach, without concern for disgrace of Yom Tov Sheni. That is because we do not count the Omer together with Kedushas Yom Tov. Here, if we would bless Leshev ba'Sukah at Kidush of Shemini Atzeres, people would disgrace Yom Tov. I say even more simply, since nowadays the calendar is fixed, mid'Oraisa we must count the Omer on the second night, so it is improper not to bless on it. Sitting in the Sukah on the eighth day is a mere custom, so we do not bless.

Question: Why didn't Chachamim enact to take the Lulav on the eighth day without blessing?

Answer #1 (Rosh, ibid.): A Lulav is Muktzeh, so taking it openly shows that he treats the day like Chol. This does not apply to Sukah, for sometimes a person enjoys eating there even on the last Yom Tov.

Korban Nesan'el (7): If it is cold or windy on Shemini Atzeres one should not eat in the Sukah.

Answer #2 (Ran, ibid.): Chachamim enacted to eat in the Sukah, for Sukah is mid'Oraisa all seven days everywhere. Outside the Mikdash, Lulav is mid'Rabanan after the first day. Also, there is no Isur to sit in a Sukah, but Tiltul Muktzeh is Asur.

Hagahos Ashri: Avi ha'Ezri says that since he does not bless like he did all seven days, this is a Heker and it is not adding to the Mitzvah. However, it is forbidden to sleep there.

Rambam (Hilchos Sukah 6:13): Nowadays that we keep two days of Yom Tov we sit in the Sukah eight days, but on the eighth day we do not bless. Similarly, Tumtum and Androginus never bless Leshev ba'Sukah for they are obligated due to Safek, and we do not bless when in Safek.

Ra'avad: We do bless on a Safek. We would bless on tithing Demai if not for the fact that the majority of Amei ha'Aretz tithe.

Lechem Mishneh: The Ra'avad holds like the Rif, that we do not bless on the eighth day because it contradicts the eighth day of Yom Tov.

Question: The Rif holds like the Rambam, that we do not bless on a Safek. Why does he say that we do not bless due to the contradiction?

Answer (Lechem Mishneh): If not for the contradiction, we would bless in order that people not be lenient and eat outside the Sukah.

Kesef Mishneh: The Rambam means that just like we sit in the Sukah on the eighth day without blessing, similarly, Tumtum and Androginus sit in it without blessing. He does not say that we do not bless due to Safek, for if so we should similarly not bless Shehecheyanu on Yom Tov Sheni! The initial decree was to bless Shehecheyanu both nights due to Safek, and our custom is to continue doing so.

Poskim

Shulchan Aruch (OC 668:1): In Chutz la'Aretz we eat in the Sukah at night and during the day but we do not bless because it is a Safek.

Beis Yosef (DH Kosav): Avi ha'Ezri forbids to sleep in the Sukah on Shemini Atzeres. The Poskim did not specify; this shows that they permit.

Darchei Moshe (2,3): The Rosh and Tur say that eating there does not show that it is Chol because sometimes a person eats in the Sukah even when it is not a Mitzvah. However, people sleep in the Sukah only for the Mitzvah. The custom is like Avi ha'Ezri.

Magen Avraham (2): We eat in the Sukah because there is a Heker that we do not add to the Mitzvah, i.e. we do not bless. We do not sleep in the Sukah because there is no Heker (we never bless on sleeping)! This is why we leave the Sukah right after eating (there is no Heker when we stay in the Sukah without eating).

Mishnah Berurah (6): Many require to sleep in the Sukah on Shemini Atzeres. The custom is to be lenient. Some say that those who forbid to sleep also forbid eating something (e.g. fruits) on which we never bless Leshev ba'Sukah, for there is no Heker. Others permit, and this is more reasonable.

Kaf ha'Chayim (6): Benei Eretz Yisrael in Chutz la'Aretz should eat in the Sukah on Shemini Atzeres, but they should explicitly say (inaudibly to others) that it is not l'Shem Mitzvah. They should find an excuse to sleep in the house, and in private eat something in the house. They should not be conspicuous, lest Benei Chutz la'Aretz learn from them to be lenient.

Beis Yosef (DH u'Mah she'Chosav v'Yesh): Some sit in the Sukah on the day of Shemini Atzeres, but not at night. This is because Yeshivas Sukah contradicts calling the day Shemini Atzeres in Kidush.

Rebuttal (Magen Avraham 2): Also during the day we call the day Shemini Atzeres in Birkas ha'Mazon! Rather, these people are concerned for Bal Tosif (adding to a Mitzvah). After ceasing to eat in the Sukah for an evening, we are not concerned for Bal Tosif. However, all the Poskim require eating all meals in the Sukah on Shemini Atzeres, even Bein ha'Shemashos at the end of the day. This is why Noyei Sukah are forbidden on day nine.

Yom Tov Sheni k'Hilchaso (2:7 and footnotes): Some say that someone from Chutz la'Aretz visiting Eretz Yisrael must sit in a Sukah on Shemini Atzeres without a Berachah (Be'er Moshe, ha'Ga'on S. ha'Levi Vozner, ha'Ga'on R. C.P. Sheinberg). Ha'Ga'on R. Y.S. Elyashiv says that one should have pots in the Sukah at the time, for then even a Ben Eretz Yisrael may sit inside. Others exempt (Lu'ach Eretz Yisrael). Ha'Ga'on R. S.Z. Auerbach exempts because even in Chutz la'Aretz there are different customs.

Maharshal (Teshuvah 68, brought in Taz 1 DH Kosav Maharshal): R. Tabil would not eat at the start of Shemini Atzeres until night, for if he would eat during the day, he would be obligated to bless Leshev ba'Sukah. The Maharshal agrees, for it is a great transgression to put oneself in a situation where he must neglect to bless. Adding from Chol to Kodesh is a Mitzvah, but it does not mean that it is now night for him. It permits to pray Ma'ariv, but he cannot fulfill Kri'as Shma or Sefiras ha'Omer until it is truly night.

Rebuttal (Taz, ibid.): If it is forbidden to eat at this time, why didn't Chachamim try to find a solution, and why didn't the Gemara or Poskim mention this Isur? The Rif, Rosh and Tur say that we bless whenever we enter the Sukah. R. Tam says that we bless on eating, which is primary, and exempt everything else. If it is forbidden to eat at this time, how can one exempt everything else? The Maharshal's reasoning should also apply to Yom Tov on Erev Shabbos. If one would begin to eat before dark he would be obligated to mention Ya'aleh v'Yavo in Birkas ha'Mazon, and this contradicts Retzei! Rather, once one adds from Chol to Kodesh like the Torah commands, it is not day for him. Nevertheless, he should wait to fulfill Kri'as Shma and Sefiras ha'Omer in the proper time. However, regarding Sukah it is like night.

Defense (Dagul me'Revavah): After praying Ma'ariv while it is still day, we are even lenient to consider it like day! (The Rema (YD 196:1) defends those who make a Hefsek Taharah to begin counting seven clean days at this time.)

Kaf ha'Chayim (10): The Acharonim refuted the Taz' proofs. Also on the ninth night we must wait until dark in order to eat in the house.